HeatEquationModel

A python model of the 2D heat equation

Usage

Under the section of code titled variables to alter, change the the variables to the way you want them to be. The section should initially look like this :

mn_level = 5 

start = -2
stop = 2.0001
step =.05 

t=1.5
dt=.003

k = .2

fn = 'Sample4'
fps = 30

#Function here:
def InitialHeatEquasion(x,y):
    return np.sin(x+y)

Each variable alters the code like so:

Function plotting

• mn_level -> the value of m and n of the fourier series (Read about it "How it works") or here
• start -> the value to start plotting the function
• stop -> the value to stop plotting the function
• step -> the step of the function plot

Time

• t -> the time to stop the simulation
• dt -> how much to step the time per frame (higher values might crash or result in odd plots, lower values slower but better)

Temprature

• a -> the thermal diffusivity of the material

Saving

• fn -> the file name
• fps -> frames per second of animation

InitialHeatEquasion

• This equation determines the initial state of the simulator must take in two parameters (x,y) and return the z coordinate which will be the temprature

How it Works

This program first does a 2D fourier approximation of the the given 2D equation. This is because the heat equation takes takes the second derivative of the function and subtracts it from the function. With a normal function this would result in a long chain of subtracted derivatives but because the second derivative of a fourier coefficent is a multiple of itself (by a decimal factor) it makes it much easier to compute. After the approximation is calculated, at the given mn_level (more on what I did in the second answer here), we then store how the function would morph with the diffusion of heat over time (in the TwoDimFourierEquasion class) and calculate it for exact points last.